Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1502 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27,23,28,21,24)(19,25,31,39,29,41,36,37)(20,26,32,43,35,44,34,38)(30,40,46,48,33,42,45,47);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(18,24)(19,29)(20,32)(21,22)(25,39)(27,28)(30,45)(33,46)(34,35)(37,41)(38,43)(40,42);(2,6)(5,7)(9,11,12,16)(10,13,14,15)(17,19,23,29)(18,25,28,41)(20,33,35,30)(21,36,22,31)(24,37,27,39)(26,42,44,40)(32,45,34,46)(38,48,43,47);(3,8)(5,7)(11,16)(13,15)(17,20,22,32,23,35,21,34)(18,26,27,43,28,44,24,38)(19,30,31,46,29,33,36,45)(25,40,39,48,41,42,37,47);(10,14)(13,15)(17,21,23,22)(18,24,28,27)(19,31,29,36)(20,34,35,32)(25,39,41,37)(26,38,44,43)(30,46,33,45)(40,48,42,47)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=h,d^2=fg,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6989820);